Preheat an oven. In a large bowl, combine chopped garlic, shredded Provolone, grated Pecorino, breadcrumbs, chopped parsley, basil, olive oil, Kosher salt, and black pepper. Stir to mix well.
Place a piece of meat in between a layer of plastic wrap, then use the toothed side of a meat mallet to pound out. Lay slice of prosciutto on each piece of meat. Divide the filling evenly between slices, spreading into an even layer on top. Roll each piece of meat tightly so that the filling is completely enclosed.
Heat neutral oil over medium-high heat in a wide, deep pan. When hot, add the braciole and cook, browning on all sides. Transfer to a plate.
Add thinly sliced garlic, then cook until fragrant and light golden brown, then deglaze with dry white wine. Bring to a boil, then use a spatula to scrape any browned bits from the bottom of the pan. Add low-sodium chicken stock, crushed tomatoes, and basil leaves, then stir.
Return the beef braciole to the sauce, along with any drippings on the plate. Cover loosely with aluminum foil, then transfer to a preheated oven and cook the meat is very tender. Baste the meat with sauce, adding a bit of water or stock.
Serve beef braciole whole with a spoonful of sauce or transfer each roll to a cutting board, thinly slice into rounds, then serve with toasted pine nuts and fresh basil or parsley.